Discussion Overview
The discussion revolves around proving that a given sequence is monotone increasing based on specific conditions provided. The participants explore the implications of the sequence's definition and the role of the supremum of the set from which the sequence is drawn.
Discussion Character
- Technical explanation, Debate/contested
Main Points Raised
- One participant presents a sequence defined by the conditions ${x}_{n-1} < x_n \le \sup S$ for all $n \ge 2$ and seeks help in proving it is monotone increasing.
- Another participant questions whether the condition ${x}_{n-1} < x_n$ already implies that the sequence is monotone increasing, assuming ${x}_{1}$ is defined.
- A subsequent reply confirms that the assumption about ${x}_{1}$ is correct and inquires if this allows one to conclude the sequence is monotone increasing.
- Another participant agrees that the strict inequality ${x}_{n-1} < x_n$ indicates strict monotonicity and suggests that the supremum does not affect the proof, questioning if there is something overlooked in the reasoning.
Areas of Agreement / Disagreement
Participants generally agree that the condition ${x}_{n-1} < x_n$ suggests the sequence is monotone increasing, but there is some uncertainty regarding the necessity of the supremum and whether additional proof is required.
Contextual Notes
The discussion does not resolve whether the supremum has any implications for the proof, and the role of ${x}_{1}$ is assumed but not explicitly defined in the initial post.